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The category of Nori motives $NMM$ is defined by tools of graph category. I think one can similarly define a category $EM$ as the graph category of $P(k)$ (the graph of projective smooth k-varieties) where the fiber functor is given by Hodge realization, is this category, after inverting $1(1)$, equivalent to the category $M(k)$ of pure motives defined by Grothendieck? (The universality of $EM$ gives the map $EM\to M(k)_{eff}$.)

Besides, has anyone proved that the category of pure motives is fully embedded into $NMM$?

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  • $\begingroup$ pure motives - There's no such thing... ;-) $\endgroup$
    – Lucian
    Commented Oct 18, 2014 at 7:58
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    $\begingroup$ What is your definition of P(k)? Do you take pairs (X,i) with morphisms induced by morphisms of varieties? About the second question: Arapura showed in arxiv.org/abs/0801.0261 that the full subcategory of pure Nori motives (in the sense of the subcategory of NMM(k) coming from smooth projective varieties) is equivalent to the category of pure André motives; see thm 6.4.1 in loc. cit. Now the category of André motives is equivalent to the category of pure motives for the numerical equivalence iff the Lefchetz standard conjecture is true for k. Relating Nori motives to cycles is hard! $\endgroup$
    – Simon Pepin Lehalleur
    Commented Nov 19, 2014 at 18:10
  • $\begingroup$ @SimonPepinLehalleur Why don't you post this as an answer? $\endgroup$
    – Leo Alonso
    Commented Nov 27, 2017 at 18:27

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